In the sammer of 1958 in St. Petersburg, Florida, a new sidewalk was poured near the childhood home of one of the authors. No expansion joints were supplied, and by mid-July the sidewalk. had been completely destroyed by thermal expansion and had to be replaced, this time with the important addition of expansion joints! This event is modeled here.
A slab of concrete $4.00$ em thick, $1.00 \mathrm{~m}$ long, and $1.00 \mathrm{~m}$ wide is poured for a sidewalk at an ambient Lemperature of $25.0^{\circ} \mathrm{C}$ and allowed to set. The slab is exposed to direct sunlight and placed in a series of such slabs without proper expansion joints, so linear expansion is prevented. (a) L'sing the linear expansion equation (Eq10.4), eliminate $\Delta l$ from the equation for compressive stress and strain (Fq. 9.3). (b) Use the expression found in part (a) to eliminate $\Delta T$ from Equation $11.3$, obtaining a symbolic equation for thermal energy transfer $Q$.
(c) Compute the mass of the concrete slab given that its density is $2.40 \times 10^{5} \mathrm{~kg} / \mathrm{m}^{3}$. (d) Concrete has an ultimate compressive strength of $2.00 \times 10^{7} \mathrm{~Pa}$, specific heat of $880 \mathrm{~J} / \mathrm{kg}+{ }^{\circ} \mathrm{C}$, and Young's modulus of $2.1 \times 10^{10} \mathrm{~Pa}$. How
much thermal energy must be transferred to the slab to reach this compressive stress? (c) What temperature change is required? (f) If the sun delivers $1.00 \times 10^{5} \mathrm{~W}$ of power to the top surface of the slab and if half the energy, on the average, is absorbed and retained, how long does it take the slab to reach the point at which it is in danger of cracking due to compressive stress?